On a Question of Bolsinov, Matveev, Miranda and Tabachnikov: Quadratic Poisson Brackets without Polynomial Integrable Systems

Bolsinov, Matveev, Miranda and Tabachnikov asked whether each quadratic Poisson bracket on a vector space admits a polynomial integrable system, that is, \(\frac12(\dim+\operatorname{corank})\) Poisson-commuting polynomials whose differentials are linearly independent almost everywhere (Question 5.9 in the numbering of arXiv:1804.03737v2; record AMR-020-0507 of the UnsolvedMath collection, where it is labelled Question 5.7); Bolsinov, Izosimov and Tsonev had asked the same question in 2017 for polynomial Poisson brackets of any degree. The answer is no. The log-canonical bracket \(\{x,y\}=xy\), \(\{y,z\}=\sqrt2\,yz\), \(\{z,x\}=\sqrt3\,zx\) on \(\mathbb{R}^3\) has generic rank two, and there are no two Poisson-commuting polynomials \(f,g\) with \(df\wedge dg\not\equiv0\). The proof is elementary: the leading exponents of commuting polynomials are orthogonal for the skew form of the bracket, and a count of dimensions does the rest; a second proof uses residues on the common level curves of \(f\) and \(g\). More generally, a log-canonical bracket \(\{x_i,x_j\}=\lambda_{ij}x_ix_j\) on \(\mathbb{R}^n\) admits a polynomial integrable system if and only if some maximal isotropic subspace of \(\lambda\) is spanned by vectors with nonnegative integer entries, and then monomials form one; in dimension three this becomes an explicit arithmetic criterion. A log-canonical bracket on \(\mathbb{R}^4\) with the constants \(1,2^{1/6},\dots,2^{5/6}\) is symplectic on a dense open set and has no polynomial integrable system, so Casimir functions are not the source of the obstruction, and multiplying the three-dimensional example by homogeneous polynomials gives counterexamples of every degree at least two. If the polynomial system is also required to contain polynomial Casimir functions, as in the formulation in the introduction of the 2018 survey, the negative answer is elementary; the substance of the result lies in the formulation of its Section 5, which does not ask for Casimir functions. The questions are due to these authors; for linear brackets the answer is positive by Sadetov's proof of the Mishchenko–Fomenko conjecture; log-canonical brackets, the bracket of two monomials and the description of the polynomial Casimir functions are classical. The contribution of this note is the counterexample with its two proofs, the characterisation for log-canonical brackets and the example on \(\mathbb{R}^4\). No printed answer to the question was found in the sources searched; a search that finds nothing is not a proof of novelty. This is an unrefereed note.